If \(E\)  and \(V\)  are Euclidean spaces then a measurable function \(g\colon E\times V\rightarrow \mathbb {R}\)  is called a Carathéodory function if \(\xi \mapsto g(x,\xi )\)  is continuous for a.e. \(y\in E\)  and \(y\mapsto g(y,\xi )\)  is measurable for every \(\xi \in V\)  (see e.g. [1]}). Given \(k\in \mathbb {N}\)  and a Carathéodory function \(g=g(x,\xi )\colon D^k\times U^k\rightarrow \mathbb {R}\)  we define the functional \(L_{g} \colon L^p(D;U) \rightarrow \mathbb {R}\)  by
\(L_{g}(u) \int _{D^k} g(x_1,\dots ,x_k,u(x_1),\dots ,u(x_k))\,dx.\) 
